Consistency and inclusion results for Toeplitz matrices of bounded linear operators

C ^OMPOSITIO M ^ATHEMATICA

B. W OOD

Consistency and inclusion results for Toeplitz matrices of bounded linear operators

Compositio Mathematica, tome 24, n

^o

3 (1972), p. 313-327

<http://www.numdam.org/item?id=CM_1972__24_3_313_0>

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313

CONSISTENCY AND INCLUSION RESULTS FOR TOEPLITZ MATRICES OF BOUNDED LINEAR OPERATORS

by B. Wood

Wolters-Noordhoff Publishing Printed in the Netherlands

1. Introduction

Recently, Ramanujan [11 ]

^{discussed some} notions of classical sum-

mability

^{in a} Banach space

setting. Summability

in certain linear topo-

logical

spaces has also been considered in

[1], [6], [9], [10], [12], [14], [15],

^and

[17]. Ramanujan’s

discussion

depends

^on

duality theory

of vector sequence spaces rather than on

regularity

conditions of the matrices involved. In this _paper we show that the

regularity

^conditions

and classical

techniques

^of

[4], [5], [7], [8], [13]

^and

[16]

^{may be}

readily employed

^to prove

consistency

and inclusion results in a

general F-space setting.

^In

particular,

^our Theorems 4.1 and 4.4 answer a

question

raised in

[11 ]

^and Theorem 5.2 extends a result of

Cowling [5].

2. Notation and Definitions

Most of our notation

corresponds

^{to that of}

[11 ].

^{Given a vector space}

E over the scalar field K of

complex numbers,

^{a vector} sequence space

S(E)

^{over E is a set} of sequences

(Xn)

^{of vectors}

^Xn

ⁱⁿ

^E,

^{which set also}

forms a linear space over under the usual

componentwise operations.

Let

w(E)

^{denote the vector} sequence space of all sequences of vectors from E. When

denoting

scalar sequence spaces, ^{we use} the standard notation of

S’,

^{instead of}

S(K).

^{In the}

sequel,

^let

(E, pi)

^and

(F, qj)

^be

F-spaces, i.e., locally

^{convex Hausdorff} spaces which are metrisable and

complete,

whose

topologies

^are

generated, respectively, by

^the countable collections

(pi)

^and

(qj)

of semi-norms. Let _{m, c, co} denote the scalar sequence spaces of

bounded,

convergent, null sequences,

respectively,

^with the sup

norm

topology.

^Denote

by lp(p

&#x3E;

0)

^the scalar sequence space of _sequences X ⁼

(Xn )

^such

that

^{oo and let}

¹⁰⁰

^{be the space of X =}

(Xn )

such that

03A3|Xn|pn

^{oo for}

^{all p}

&#x3E; 0. The _space

lp

^{has the norm}

^topology

given by ~X~

⁼

,

^while

¹⁰⁰

^{has the}

F-topology given by

^the

family

{hn:n

⁼

1, 2, ···} of semi-norms, with hn(X)

⁼

max {|03A3Xkzk| : ^{|z| =} n}.

We shall be concerned with the

following

^vector sequence spaces:

Let A ⁼

(Ank)

^{denote an infinite matrix of} bounded linear operators of E into F. Given X in

w(E), formally

^define y ⁼

(Yn) by

and write _y ⁼ AX. If

a(E), 03B2(F)

^{are two vector sequence spaces, we say}

that A ^E

r( a(E), 03B2(F))

^{if X E}

a(F) ^implies

^{y E}

03B2(F),

^{Call A reversible}

if the

equation

y ⁼ AX has a

unique

solution X in

a(E)

^{for each y in}

03B2(F).

^Denote

by DA(E, F),

^{the domain}

of A,

^{the set}

{X

⁼

(Xn)

^E

w(E):

^{y = AX exists in}

w(F)}.

^The

p-summability field of A,

^denoted

by 03B2A(E, F)

^{is the set}

{X

⁼

(Xn)

^E

w(E) :

^{y = AX} exists in

w(F)

^and

y ^E

03B2(F)}.

3. Preliminaries

In this section we shall state a number of results for the

subsequent consistency

^and inclusion theorems.

PROPOSITION 3.1

[10]. If E,

^{F are}

locally

^{convex spaces and E is}

quasi- complete (closed

^{and bounded sets}

of

^{E are}

complete)

^{then any collection}

of

continuous linear operators

from

^{E into F which is}

simply

^{bounded is}

bounded for

^the

topology of uniform

convergence ^on bounded sets.

Note that if E is ^an

F-space

^{it is}

quasi-complete

^and also barrelled.

PROPOSITION 3.2

[10].

^Let

(Tn)

^{be a sequence}

of

continuous linear operators ^{on E into} F, ^where

E,

^{F are}

F-spaces. If limn Tn(X) ^exists.for

each X in

a fundamental

^set

of E

^and

if for

^{each X in}

E, (Tn(X))

^{is bounded}

in F then

T(X)

⁼

limn Tn(X)

^exists

^for

^{all X in}

^E,

^{and T is a continuous}

linear operator

of

^{E into F.}

The next two

propositions

^are

easily proved using

^the

techniques

^of

[10], [14]

^and

[15].

PROPOSITION 3.3 The matrix A ⁼

(Ank)

^E

r(co(E), c(F)) ^if

^and

only if

(i) for

each bounded set

Ma

^{in E}

and for each fixedj,

and Xk ~ M03B1, k = 0, 1, ···, and

(ii) for

each X E E and each

fixed

^{k =}

0, 1, ···, limn AnkX

^exists.

Also,

^lim

AX

^~

limn Lk ^AnkXk

^{= 0} whenever lim X ⁼ 0

if

^and

only if

(i)

^{holds and}

(ii)

^is

specialized

^to

(ii)’ ^{limn AnkX}

^{= 0}

^for

^{each XE E and each k =}

^{0, 1,...}

PROPOSITION 3.4. The matrix A ⁼

(Ank)

^E

0393(l1 (E), c(F)) ^if

^and

^{only if} (i) ^for

^{each bounded set}

^Ma

^{in E}

and for each fixedj,

and Xk ~ M03B1, k = 0, 1, ···; and

(ii) for

^{each X E E and}

each fixed

^{k =}

0, 1, ···, limn AnkX

^exists.

If (i)

and

(ii)

hold then lim

AX

⁼

Lk ^{limn AnkXk for}

^{X E}

11(E’).

PROPOSITION 3.5. Let _p and _q be

positive

^numbers.

(i)

^{A =}

(Ank)

^E

r(lp(E), lq(F)) ^if

^and

only if for

^{each bounded set}

Ma

^{in E}

and for

^each

fixed j,

and Xk ~ M03B1, k = 0, 1, ···.

(ii) If, for

^each

positive integer

^{r there exists an s =}

s(r) ~

^{r such that}

for

^{each bounded set}

Ma

^{in E}

and for

^each

fixed j

⁼

1, 2, ...,

and

Xk

^E

M03B1,

^{k =}

0, 1, ...,

^{then A =}

(Ank)

^E

F(1.(E), l~(F)).

^{The con-}

verse holds

if

^the

topology of

^{F is}

given by

^the

finite

collection

(qi),

^{i =}

1,

···,

1.

(iii) If

^{there exists an s =}

s(q)

&#x3E; 0 such that

for

each bounded set

M(1.

^{in E}

and for

^each

fixed j

⁼

1, 2, ...,

and

Xk

^E

Ma,

^{k =}

0, 1, ...,

^{then A =}

(Ank)

^E

r(loo(E), 1,,(F».

^{The con-}

verse holds

if

^the

topology of

^{F is} given

by

^the

finite

collection

(qi)’

^{i =}

1, ...,

^1.

(iv)

^{A =}

(Ank)

^E

r(lp(E), l~(F)) if and only if for

^{all r} &#x3E; 0

and for

^each

bounded set

Ma

^{in E}

and for

^each

fixed j

⁼

1, 2, ...,

and XkEM(1.,k

⁼

0, 1,....

PROOF. Part

(i)

follows from

[15]

^and the fact that the _maps

(Xk) H (Xkpk)

^and

(yn)

^H

(ynpn)

^are one-to-one

correspondences

^between

lp(E)

and

l1(E)

^{and between}

IP(F)

^and

h (F), respectively.

We prove

(ii)

^{as follows.}

Suppose

^{that for each}

positive integer r

^there

exists an s ⁼

s(r) ~ r

^{such that A E}

F(1,(E), lr(F)),

^{i.e. the condition}

of

(ii)

^{holds. We note that}

lv(E) ~ lw(E)

^{if 0 w v and}

h (E) -

~~v=1 ^lv(E).

^{Let X =}

(Xn)

^E

1. (E)

^{and let a}

positive integer r

^be

given.

Therefore,

there exists an s ⁼

s(r) ~ r

^{such that A E}

r(ls(E), lr(F)).

Since

XE Is(E), A(X)~lr(F).

This shows that A ^E

0393(l~(E), l~(F)).

Conversely,

^{assume the}

topology

^{of F is}

given by

the finite collection

(qi)i=1

and let A ^E

0393(l~(E), l~(F)).

For each X E

lp(E),

^let

03C3ip(X) = n=

⁰

Pi(Xn)pn,

p &#x3E;

0,

^{i =}

1, 2, ....

^Then

03C3ip

^{is a seminorm on}

Ip(E).

Let the

topology

^of

1,,,,(E)be ^{given by}

^{the sup of}

0(vO)

^{where 0}

{03C3ip :

^{p =}

1, 2,... ;

^{i =}

1, 2, ···}.

^Then

(l~(E), v4»

^{is a}

locally

^convex

complete

space.

Let q

^{be a}

given positive integer .Then

^{A E}

0393(l~(E), lq(F))

and for each _y ⁼

(yn)

^E

lq(F)

^{we let}

uq(y) = 03A3~n=0 t(Yn)qn,

^where

t ⁼

.

^Then

(lq(F), u.)

^{is a}

locally

^convex

complete

space and A is continuous as a map from

(l~(E), v03A6)

^to

(1,,(F), uq).

Thus there exists

a number M and a finite

collection,

say

03C311, ···, 03C3ip,

of seminorms such that for each X ~

1,,,, (E)

Since,

^for

each j, 03C3jr(X) ~ 03C3js(X) ^{if r ~ s,}

^{we may choose p such that}

p ~ q

^and

Let

a(X)

⁼

03C31p(X)+···+03C3ip(X)

^and

(lp(E), a)

^{is a}

locally

^{convex com-}

plete

space. Since

11(E)!;2 l2(E) ~···~ lp(E)

^{it follows that A is}

continuous as a map from

l~(E)

^{with the}

a-topology

^of

lp(E)

^to

(lq(F), uq).

For vectors t~E define b"t ⁼

(0,0,...,

^t,

03B8, ···) (0

^{is the}

zero

vector).

^Then

03B4nt ~ l~(E), n

⁼

0, 1, ···,

and the series

03A3

⁰

^03B4nXn

converges to X ⁼

(Xn)

^{in the}

a-topology

for each X E

lp(E).

^Therefore

l~(E)

is dense in

(1,(E), a)

^{and we may extend A to a} continuous linear operator ^{T from}

(lp(E), ^a)

^to

(1,(F), uq).

^{But for X =}

(Xk)

^e

IP(E),

and so for each n ⁼

0, 1, ···,

which shows that T is still

given by

^{the matrix A.}

Therefore,

^{to each}

q ⁼

1, 2, ···,

there exists ^a _p ⁼

p(q) ~ q

^{such that A E}

0393(lp(E), lq(F)).

The

proof

^of

(iii)

^{is similar to that of}

(ii)

^and

(iv)

^{follows from the fact}

that

1. (F)

^{= 0}

1,(F).

Let

w(E), w(F)

^be

given

^the weakest linear

topologies

^{such that the}

coordinates are

continuous, i.e., w(E)

^{has the}

F-topology given by

^the

family

of semi-norms

03C0ki(X)

⁼

pi(Xk), k

⁼

0, 1, ..., i = 1, 2, ...

^and

w(F)

^{has the}

F topology given by

the collection

(Jkj(Y)

⁼

qi(Yk)

k ⁼

0, 1, ···, j = 1, 2, ···.

^{If E is an}

F-space

^{and H =}

w(E)

^{has its}

topology given

^as

^above,

^{then the}

corresponding FH-spaces (see [13])

will be called

FK-spaces

^{over E.}

PROPOSITION 3.6. Let 03BB be _any one

ofc,co,lp,loo.

^Then

DA(E, F)

^and

ÂA(E, F)

^are

FK-spaces

^over

E,

^with semi-norms

{03C0ki, f3kj :

^{k =}

^{0, 1, ···,}

i, j, =

1,

2, ···}

^and

{03C0ki, 03B2kj, : k, n = 0, 1, ···, i, j = 1,,2, ···},

^re-

spectively, where, for

^{X =}

(Xn)

^E

D A(E, F)

and, for

^{X E}

03BBA(E, F), i:(X)

⁼

in(AX)

^where

^~n

^{are the semi-norms}

generating

^the

F-topology of 03BB(F).

If

^{A is}

row-finite f3kj

^{may be}

^{omitted; if A}

^is

reversible,

7tki,

f3kj

^{may be}

omitted.

PROOF. The result follows from

Proposition

^3.2 and standard techni- ques found in

[ 13,

p.

227].

It is easy ^to show that various choices of À lead to the

following generating

^{families for the}

F-topology

^of

03BB(F):

4.

Consistency

^Theorems

Foi convenience we shall call the

pair (a, j8)

admissible if a is one of

l p , ^{100 where p ~ 1,}

^and

^fl

^{is one of}

lq , 100 where q ~

^{1. If a} is any ^one of c, co,

lp, 100 and p

^is any ^one of _{c, co,}

lq, ^l~

^{and A E}

r( a(E), 03B2(F)),

^we

say that A

is perfect

^if

03B1(E)

^{is dense in}

03B2A(E, F),

^where

03B2A(E, F)

^{has the}

F-topology

^of

Proposition

^{3.6. Let}

(a, ^03B2)

be admissible and

A,

^{B be in}

0393(03B1(E), 03B2(F)).

^{For X E}

03B2A(E, F)

^{define 03C3 o}

A(X) = ¿LnkAnkXk.

^We

write B - A if 03C3 o

A(X) = (j

^o

B(X)

^{for X E}

a(E).

^{If 03C3 o}

A(X) = 03C3

^o

B( X)

for X E

03B2B(E, F)

ⁿ

PA (E, F),

^{we say that B} is (j-consistent with A.

THEOREM 4.1. Let

(03B1, 03B2)

be admissible and A

~ 0393(03B1(E), 03B2(F)). ^If

a ⁼

03B2 = 100

^{or a =}

l~

^{we assume that the}

topology of

^{F is}

given by

^the

qi, i ⁼ 1... 1. Then ^03C3 o A E

L(P,(E, F), F) (continuous

linear operators

on

03B2A(E, F)

^to

F)

^{and A} is 03C3-consistent with _every B E

0393(03B1(E), 03B2(F))

^such

that B - A and

03B2A(E, F) ~ 03B2B(E, F) ^if

^and

^{only if}

^{A is}

^perfect.

We shall

give

^the

proof

^{for the a =}

fi

⁼

100

^{case. The other cases will}

be seen to be similar. First we

require

^{two lemmas on the}

representation

of continuous linear functionals.

LEMMA 4.2.

If

p ^E

[1.,(E, F)]’ (topological

^dual

of l~A(E’, F))

^then

there exist _g E

[DA(E, F)]’,

^{h E}

[loo(F)]’

^such

that, for

^{X E}

l~A(E, F),

The inverse

projections /1;1 E~DA(E,F) and 03C0-1n: F ~ l~(F)

^are

defined by 03BC-1k(t) = ()k t, 03C0-103C0(t)

⁼

^03B4nt,

^{where 03B4nt =}

(0, 03B8, 03B8, ···,

t,

0, ...).

^We

have g~03BC-1k~E’

^and

PROOF.

If 03C1~[l~A(E’, F)]’ = (A-1[l~(F)])’,

^a version of Theorem 5 of

[13,

p.

230]

^{for vector} sequence spaces

implies

^there

exist g

^E

[DA(E, F)]’

^{and h e}

[l~(F)]’

^{such that for all X E}

l~A(E, F), p(X) = g(X) + h~A(X).

^If

X = (Xk)~DA(E’, F)

^then

0, ...)

^and

^clearly

^{in the}

DA(E, F)-topology.

^Therefore

g(X) = g(03B4kXk),

the series

converging

^{in K.}

Similarly, if y

⁼

(Yn) E l~(F)

^then

03B4kyk ~ y

^{in the}

topology of l~(F)

^{and we obtain}

h(y) = h(03B4kyk)

^where the series of scalars converges. If

,uk 1, n; 1

^{are defined as in the statement of the}

lemma,

^then

Proposition

^3.6

shows that

03BC-1k~L(E, DA(E, F))

^and

~L(F, l~(F)). ^Thus,

^for

X ⁼

(Xk )

^e

l~A(E, F) and y

⁼

(Yn)

⁼

^Ax,

LEMMA 4.3.

If t

^e

(F,

^{q i , i =}

1, ···, 1)

^{and p e}

[l~A(E, F)]’

^{then there}

exists a method

B = B(t)

^{such that}

B~0393(l~(E), l~(F)), l~A(E, F)

~

l~B(E, F)

^{and fI o}

B(X)

⁼

p(X)t

for all X E

l~A(E, F).

PROOF. In the notation of Lemma

4.2,

^{define B =}

(Bnk) by

^the

following equations.

^For

X ~ E,

^let

Clearly, Bnk~L(E,F) and 03C3~B(X) = 03C1(X)t

^for

X ~ l~A(E, F).

^We

apply Proposition

^3.5

(ii)

^to show that B E

h(h (F), l~(F)).

Fix j

^{and a bounded set}

Ma

^{in E. Let X =}

(Xk)

^where

^Xk

^e

^Mx (k

⁼

^0, 1, ...)

^and

^{let q}

^{be a}

positive integer.

^{We must show the exis-}

tence

of p o

⁼

p0(q) ~ q

^{and a number K =}

K03B1,j,p0,g

^{such that}

Let k be a

given positive integer. Then,

for any

positive integer

p,

We have

and we shall show the existence of a

positive integer

pi and ^a number R ⁼

R(a, pl)

^{such that}

Let z,

n, i, y

^be

given positive integers

^and

suppose pi(Xk) ~ ^1,

k ⁼

0, 1,...

It follows that

and

Since A E

F(1,,,,,(E), l~(F)),

^for

each 03BC

⁼

1, 2, ...

there exists a

positive

integer

^{z =}

z(y)

^{and a number R’ =}

R’(03BC)

^{such that}

Since g

^E

[DA(E, F)]’,

there exists a number M ? ^{0 and}

positive integers

a,

b, c, d

^such

that,

for any

positive integer

p,

for k ⁼

0, 1, ···.

^It follows that there exists a

positive integer

pl and number R ⁼

R(03B1,p1)

^{such that}

Next,

^{since A}

~ 0393(l~(E), 1.(F», Proposition

^3.5

(ii) yields

^{for each}

positive integer

^{w, an s =}

s(w) ~ w

^{and a number T =}

T(a,

^{p, s,}

w)

such that

We have

03C0-1n

^o

Ank(Xk) = c5nAnk(Xk) = (03B8, 03B8,

^...,

Ank’ 03B8, ...)

^with

AnkXk appearing

^{in the n-th}

position.

^Let

yk = (yku) with yku = 03B8(u ~ n), ykn = AnkXk, k

⁼

0, 1, ....

The remark

following Proposition

3.6 says that the

toplogy

^of

l~(F)

^is

^{given by}

^the

collection {Q03BC,r :

^{03BC, r =}

1, 2, ···}

with

Since h E

[l~(F)]’,

^{there exists a number}

^{N ~}

^{0 and}

positive integers e, f

^{such that}

It follows

that,

^for

positive integers

p,

Thus there exists an si =

s1(q) ~ q

^{and a}

number Ti = Tl (a,j,

^si,

q)

such that

Finally, combining

^{the above we obtain} a p o

= p0(q) ~ q

^{and a number}

K ⁼

K(03B1, j, p0, q)

^{such that}

(4.1)

holds. The

proof

^that

l~A(E, F) ~ l~B(E, F)

^is

straightforward

^and Lemma 4.3 is

proved.

PROOF oF THEOREM 4.1.

Repeated applications

^of

Proposition

^3.2

show

that 03C3~A ~ L(l~A(E, F), F) ^and,

^if

100 A (E, F) ~ lOOB(E, F),

6 o B E

L(1,,,,(E, F), F)

^also.

^Utilizing

^{Lemma 4.3}

^above,

^the

proof

^now

goes

through just

^{like that of}

[4,

^Theorem

3 ].

Let

A, B ~ F(co(E), co(F)).

^Iflim

^BX

^{= 0 for X ~}

coA(E, F)

ⁿ

cB(E, F)

we say that A is consistent with B. Thus

coA(E, F) ~ cB(E, F)

^{and A}

consistent with B

imply co,(E, F) ~ coB(E, F).

THEOREM 4.4 Let A E

0393(c0(F), c0(F)).

(i) If X is in

^the

co,(E, F)-closure of co(E)

^{then lim}

^BX

^{= 0}

^for

^every

B E

r(co(E), co(F))

^{such that}

coA(E, F) ~ cB(E, F).

(ii) If X E c0A(E, F)

^{is not in the} coA

(E, F)-closure of co(E) then, for

any scalar _Il and non-zero t E F, ^{there exists a B E}

r(co(E), c0(F))

^such

that

c0A(E, F) 9 cB(E, F)

^and

^{lim BX}

^{= j1t.}

(iii)

^{Method A} is consistent with _every

B ~ T(c0(E), co(F))

^{such that}

c0A(E, F) 9 ^{cB (E,} F) if and only if A

^is

perfect.

PROOF.

Clearly, (iii)

^{follows from}

(i)

^and

(ii).

^Part

(i)

is obvious since

coA(E, F) ~ cB(E, F) ^implies (by Proposition 3.2)

^that

^limB

is continuous

on

coA(E, F)

^{in the}

^topology

^{of that space. Under the}

hypotheses

^of

(ii)

we

choose f~ [coA(E, F)]’

^such

that f(X) =

^j1

and f vanishes

^on

co (E).

Then we construct a B E

r(co(E), c0(F))

^{such that}

coA(E, F) ~ cB(E, ^F)

and

limB y

⁼

f(y)t

^for

^{all y}

^E

coA(E, F). First,

^since

03A3mk=

⁰

^03B4kzk

^{~ z in the}

topology

^of

c0(F)

^{if z E}

c0(F)

^{we have}

(in

^{a manner similar to the}

^proof

of Lemma

4.2)

for y

⁼

(yk)

^E

c0A(E, F),

^where

Il; 1, nn ’

^{are as} in Lemma 4.2 with

1,,,,(F) replaced by co(F),

^{g E}

[DA(E, F)]’,

^{and h E}

[co(F)]’.

^Define

B ⁼

(Bmk)by

^the

equations

We have

for _y ⁼

(y,)

^E

cOA(E, F)

^{and m =}

0, 1,2, ..., since,

^{for each} n,

03A3~k=0h

^{~03C0-1 o}

A nk(Yk)

^{is a} convergent series of scalars.

Therefore,

It follows that B e

r(co(E), co(F))

^and

coA(E, F) ~ cB(E, F).

^This

completes

^the

proof

^{of Theorem 4.4.}

A sequence

(fn)

of continuous linear functionals on an

espace

^{F is} said to

satisfy property P(A, 03B2)

^{if it is bounded for the}

topology

^{of uni-}

form convergence ^on bounded sets and

knfn(AnkXk)

converges in K for

every X = (Xk)Ef3A(E,F)

^{where A =}

(Ank) ~ 0393(03B1(E), 03B2(F))

^and

(a, 03B2)

is admissible.

THEOREM 4.5. Let

(a, 13)

^be admissible and A

~ 0393(03B1(E), f3(F»). If

a

= 13

⁼

l~

^{or ri =}

l~

^{let the}

topology of Fbe given by

qi, i =

1, ..., 1.

Then A

is perfect if and only if for

^each

(fn) satisfying P(A, 13)

^{we have}

for all X ⁼

(Xk)

^E

03B2A(E, F).

PROOF. We

give

^the

proof

^{for the a}

= 03B2

⁼

100

^{case. It follows from}

Lemma 4.2 and

Proposition

^3.1

that f~ [l~A(E, F)]’ ^implies

^{there exist}

uniformly

^bounded sequences

(gn), (hn) ^{with gn}

^E

^{E’, hn}

^{E F’ such that}

for all X ⁼

(X,)

^E

looA(E, F).

^The

^proof

^now follows the lines of

[4,

Theorem

1 ].

In

[2]

^Brown

gives

^an

example

^{of a}

perfect

non-reversible

Il - 11

matrix A ⁼

(ank)

^{of scalars for which}

diverges

^{for a certain}

XE 11A

^and

^uniformly

^{bounded sequence}

(f,)

^of

continuous linear functionals. In his

example fn(X)

^{= X} for all X e

K, n = 0, 1, ···,

Let

(a, fi)

^be admissible and A ⁼

(Ank)

^e

r(a(E), P(F».

^{We say that}

A is type ^M*

if,

for every sequence

(fn)

^of continuous linear functionals

on F which is bounded for the

topology

of uniform convergence ^on bounded sets, the conditions

imply fn ~ 0, n = 0, 1, ···.

THEOREM 4.6. Let

(a, fi)

^be admissible and A E

r( a(E), 03B2(F))

^be

reversible.

If

^{a =}

13

⁼

l~

^{or a =}

l~

^let

(qi) = (qi,

^{i =}

^{1, ···l).}

^{Then A}

is

perfect if

^and

only if

^{A is} type ^M*.

PROOF.

Using

^the

representation

^of

[03B2A(E, F)]’ given

^{in the}

proof

of Theorem

4.5,

^the

proof becomes

^a

repetition

^{of that of}

[4,

^Theorem

2].

We _may now state the

following corollary

which extends a result of

Macphail [7].

^{The case a =}

fi

⁼

l~

^{is not} considered in

[7].

COROLLARY 4.7. Let

(a, 03B2)

be admissible and A e

r(a(E), 03B2(F))

^be

reversible.

If

^{a =}

fi

⁼

l~

^{or oc =}

l~

^let

(qi) = (qi,

^{i = 1...}

1).

^{Then A is}

a-consistent with _{every B E}

0393(03B1(E), 03B2(F))

^{such that B - A and}

13 A(E, F)

g

f3B(E, F) ^if

^and

^{only if}

^{A is type M*.}

Let

(a, fi)

be admissible and A ⁼

(Ank) ~ 0393(03B1(E), 03B2(F)).

^{An X =}

(Xk)

^E

03B2A(E, F)

^{is said to be}

^perfect

^if

for all sequences

(ln)

of continuous linear functionals on F which

satisfy P(A, 03B2).

^{An X =}

(X,)

^e

PA(E, F)

^{is said to be type N if}

for all

(fn) satisfying P(A, 03B2).

PROPOSITION 4.8. Let

(a, fi)

be admissible and A e

r(a(E), 03B2(F)). ^If 03B1 = 03B2 = l~ or 03B1 = l~ let (qi) = (qi, i = 1, ···, l).

(i)

^{X e}

03B2A(E, F) is perfect if and only if Xe 03B2A(E.F)-closure ^of 03B1(F);

(ii) ^{if X ~ 03B2A(E,} F)

^{and A is} reversible then X

is perfect if X is

type ^N.

PROOF. The

proof

^of

(i)

^{follows the lines of}

^[2,

^Lemma

^2].

(ii)

^Let

X~03B2A(E, F)

^be

type N

^and

/e [03B2A(E, F)]’, f(y)

^{= 0 for}

y ^e

a(E).

It follows from the

representation

^of

[PA(E, F)]’

^that

where

(hn)

^satisfies

P(A, /3).

^{Since A is}

reversible, Proposition

^3.6 says that the

topology

^of

/3A(E, F)

^is

given by ()

^where

(y)

⁼

in(Ay)

and

(in)

^{generates the}

topology

^of

/3(F).

^{It follows}

^easily

^that

where

(AJ

^satisfies

P(A, fl).

^Therefore

f(X)

^{= 0 and X is}

^perfect.

The

following example, suggested by

^the

referee,

^{shows that the con-}

verse of 4.8

(ii)

^is false. Take

03B1 = 03B2 = l1, E = F = K,

^{and A =}

identity

matrix. Then

A ~ 0393(l1, l1),

^each

XE 11A

^is

^perfect,

^{A is}

reversible,

yet every

X ~ l1A

^{is not type N.}

Let

(a, 03B2)

be admissible and A E

r( a(E), P(F)

^be reversible. If

= fi

⁼

l~

^{or a =}

l~

^let

(qi) _ (qi, i

⁼

1, ... 1).

Consider the follow-

ing :

(i)

^{A is type}

^M*;

(ii)

^{A is}

^perfect;

(iii)

^{all X E}

03B2A(E, F)

^are

^perfect;

(iv)

^{all X e}

03B2A(E, F)

^{are type N.}

Then

(i)

^~

(ii)

^~

(iii)

^~

(iv).

The next result

gives

^{the usual} characterization of type ^{M* for our}

setting.

THEOREM 4.9. Let

(a, fi)

be admissible and A E

0393(03B1(E’), 03B2(F)). ^If

a

= 03B2

⁼

l~

^{or a =}

l~

^let

(qi) = (qi, i

^{= 1,...}

^l).

^The

^following

^are

equivalent:

(i)

^{A is type}

^M*;

(ii)

^The collection

of

sequences

(A,,k (X»,

^{k =}

0, 1, ···,

^{X E}

E,

^is

fundamental

ⁱⁿ

03B2(F);

(iii) A[03B1(E)] is fundamental

ⁱⁿ

03B2(F).

PROOF. The

proof employs

^known

procedures (see [11])

^{and the}

following

^facts.

Let

(f.)

^{be a}

uniformly

^bounded sequence of continuous linear functionals on F and define

f(y) = fn(yn), y~ ^P(F).

^{We claim that}

f e (j8(F))’. (The proof

^is

given for

⁼

11;

^{the other cases are}

similar).

Let y

⁼

(Yn)

^e

l1(F)

^and

rn(X) = rh(X)r |

^{for X E F.}

Then rn

^{is a semi-}

norm on F

and,

^since

(f")

is bounded for the

topology

of uniform conver-

gence ^on bounded sets, there ^{exists a} number M

= M(j) ~ ^{0, independ-}

ent of _n, such that

rn(X) ~ Mqj(X), j

⁼

^1,

^{2, ..., n =}

0, 1, .... (See,

e.g.,

[14]

^or

[15]).

^Therefore

and

f E (ll(F))’.

^{Now let z E}

A[03B1(E)]

^{and write z =}

A(03BC), 03BC

^E

a(E).

We claim that

We

verify

^{this as follows for the a =}

fl = h

^{case. If A =}

(Ank)

^E

0393(l1(E), l1(F))

^and

( fn)

is bounded for the

topology

of uniform conver- gence ^on bounded

sets, fn

^E

F’,

^we claim that B ⁼

( fn

^o

Ank)

^E

0393(l1 (E), 11).

Clearly fn

^o

Ank

^{E E’ for all}

^{n, k}

^{and we use}

Proposition

^3.5

(i)

^{to show}

that B ~ 0393(l1(E), l1).

^Let

^Xu

^E

Mq ,

^{a bounded set in}

^E,

^{for u =}

^0,1,....

Since

(fn)

^{is bounded for the}

^topology

of uniform convergence ^on bound- ed

sets, just

^{as before we}

have, for j

⁼

1, 2, ···,

^a

number M(j) ~

⁰

such that

Proposition

^3.5

(i) ^implies

^{that B~}

0393(l1 (E), 11).

^The

required interchange

of summation now follows from observations in

[15].

5. Inclusion Theorems

Suppose ri, f3

^are any of c, co,

lp, ¹⁰⁰

^{and A =}

(Ank)

^E

r( a(E), 03B2(F))

is reversible. Fix X E F and let the

sequence 03BEp

⁼

03BEp(X) correspond

^to

ôPX under

A, i.e.,

Define maps

!p,n

^{from F to E}

^by

^the

^equations fp,n(X) = ^03BEpn,

^{n, p =}

0,1, ···. Clearly, f p, n ~ L(F, E)

^{for all n, p.}

THEOREM 5.1. Let A

E r(co(E), co(F))

^be reversible and

let fp, n,

^{n, p =}

0, 1, ···

^{be the} maps

defined

^{above. Let B =}

(Bnk)

^{be a row}

finite

^matrix

of

continuous linear operators ^{on E to F.} Then

c OA (E, F) 9 cB(E, F) if

and

only if

PROOF. If X ⁼

(X,)

^E

coA(E, F) ^{and y}

^{= AX we define}

Pk(Y)

⁼

^Xk.

Using

the fact that A is reversible it is easy ^to show

that pk

^E

L(co(F), E),

k ⁼

0, 1, ···.

^It follows that

where _pnk E

L(F, E).

^{We can}

easily

^{show that} 03C1nk ~

fkn

^{for all}

^{n, k}

^{so that}

Then

and the series on the left converges in F whenever

(Xk)

^E

coA(E, F)

^since

B is row finite.

Using

the fact that B is row

finite,

^{we then have}

Finally,

the left-hand sequence is in

c(F)

^{if and}

only

^if

(i)

^holds.

We have an

analogous

^{result for absolute}

summability (see

^also

[5]).

THEOREM 5.2. Let A E

0393(l1(E), 11(F)

^be reversible and let B ⁼

(Bnk)

be a matrix

of

continuous linear operators ^{on E to F. Then}

11A(E, F) z 11B(E, F) if

^and

only if

The author is indebted to the referee for a number of

helpful

sug-

gestions

^{and to} Professor H. I. Brown for

helpful

^comments

concerning

the

proof of Proposition

^3.5

(ii).

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[1] ] Consistency theorems for Banach space analogues ^of Toeplitzian ^{methods of} summability, ^Studia Math., ^XVIII (1959), ^199-210.

H. I. BROWN

[2] The summability ^{field of a} perfect l-l method of summation, ^Journal D’Analy.

Math., ^XX (1967), ^281-287.

H. I. BROWN

[3] Entire methods of summation, Comp. ^{Math. 21} (1969), ^35-42.

H. I. BROWN AND V. F. COWLING

[4] ^On consistency of l-l methods of summation, Mich. Math. J. 12 (1965), ^357-362.

V. F. COWLING

[5] Inclusion relations between matrices, Math. Zeitschr. 98 (1967), ^192-195.

R. H. COX AND R. E. POWELL

[6] Regularity ^{of net} summability transforms on certain linear topological spaces, Proc. A.M.S. 21 No. 2 (1969), ^471-477.

M. S. MACPHAIL

[7] Some theorems on absolute summability, Canad. J. Math. 3 (1951), ^386-390.

S. MAZUR

[8] Eine Anwendung der Theorie der Operationen ^{bei der} Untersuchung ^der Toeplitzchen Limitierungsverfahren, Studia Math. 2 (1930), 40-50.

H. MELVIN - MELVIN

[9] Generalized K-transformations in Banach spaces, Proc. London Math. Soc.

(second series) ⁵³ (1951), ^83-108.

M. S. RAMANUJAN

[10] Generalized Kojima-Toeplitz matrices in certain linear topological spaces, Math.

Annalen 159 (1965), ^365-373.

M. S. RAMANUJAN

[11] Vector sequence spaces and perfect summability matrices of operators ^{in Banach} spaces, to appear in Publ. of the Ramanujan Inst. India.

A. ROBINSON

[12] ^On functional transformations and summability, ^Proc. London Math. Soc.

(second series) ⁵² (1950), ^132-160.

A. WILANSKY

[13] Functional Analysis, Blaisdell, ^New York, ^1964.

B. WOOD

[14] ^{Series to} sequence and series to series transformations in Fréchet spaces, Math.

Annalen 184 (1970), ^224-232.

B. WOOD

[15] ^{On l-l} summability, Proc. A.M.S. Vol. 25 2 June 1970, p. 433-437.

K. ZELLER

[16] Allgemeine Eigenschaften ^von Limitierungsverfahren, ^{Math. Z. 53} (1951), 463-487.

K. ZELLER

[17] Verallgemeinerte Matrixtransformationen, ^{Math. Z. 56} (1952), ^18-20.

(Oblatum 22-VII-70) ^{Prof. B.} Wood,

Department ^of Mathematics, The University ^of Arizona, Tucson, Arizona, ⁸⁵⁷²¹ U.S.A.